带类时边界的整体双曲流形上Friedrichs系统的柯西问题研究

The initial boundary value problem for the Einstein equations with totally geodesic timelike boundary

We prove the well-posedness of the initial boundary value problem for the Einstein equations with sole boundary condition the requirement that the timelike boundary is totally geodesic. This provides the first well-posedness result for this specific geometric boundary condition and the first setting for which geometric uniqueness in the original sense of Friedrich holds for the initial boundary value problem. Our proof relies on the ADM system for the Einstein vacuum equations, formulated with respect to a parallelly propagated orthonormal frame along timelike geodesics. As an independent resu…

带有APS边界的洛伦兹Dirac算子的柯西问题

The Cauchy problem of the Lorentzian Dirac operator with APS boundary conditions

本文研究了带有APS边界条件的洛伦兹Dirac算子的柯西问题，通过能量估计和光滑算子分析，证明了初边值问题的适定性及解的光滑性。

粘性旋转浅水方程的Cauchy问题

Cauchy problem for viscous rotating shallow water equations

本文考虑粘性可压缩旋转浅水系统的Cacuhy问题，该问题是最近在旋转子域中自由表面浅水运动模拟中导出的，其中包含三阶表面张力项。当初始数据接近远离真空的恒定平衡态时，证明了Besov型空间中解的整体存在性。与先前关于无科里奥利力的可压缩流体模型的分析不同，旋转效应引起了Hodge速度向量场分解的两个部分之间的耦合，需要额外的正则性以执行Friedrichs正则化和紧致性论证。

The Cauchy problem for a quasilinear system of equations with coupling in the linearization

The Cauchy problem for a quasilinear system of hyperbolic-parabolic equations is addressed with the method of linearization and fixed point. Coupling between the hyperbolic and parabolic variables is allowed in the linearization and we do not assume the Friedrich's symmetrizability of the system. This coupling results in linear energy estimates that prevent the application of Banach's contraction principle. A metric fixed point theorem is developed in order to conclude the local existence and uniqueness of solutions. We show that the boundedness in the high norm and contraction in the low norm…

Global solutions and Relaxation Limit to the Cauchy Problem of a Hydrodynamic Model for Semiconductors

It is well-known that due to the lack of a technique to obtain the a-priori $L^{\infty}$ estimate of the artificial viscosity solutions of the Cauchy problem for the one-dimensional Euler-Poisson (or hydrodynamic) model for semiconductors, where the energy equation is replaced by a pressure-density relation, over the past three decades, all solutions of this model were obtained by using the Lax-Friedrichs, Godounov schemes and Glimm scheme for both the initial-boundary value problem \cite{Zh1,Li} and the Cauchy problem \cite{MN1,PRV,HLY}; or by using the vanishing artificial viscosity method f…

Global well-posedness for the full compressible Navier-Stokes equations

In this paper, we mainly study the Cauchy problem for the full compressible Navier-Stokes equations in Sobolev spaces. We establish the global well-posedness of the equations with small initial data by using Friedrich's method and compactness arguments.

Global small solution and optimal decay rate for the Korteweg system in Besov spaces

In this paper we consider the Cauchy problem to the Korteweg system with the general pressure in dimension $d\geq2$, and establish the global well-posedness of strong solution for the small initial data in $L^p$ type critical Besov spaces by using the Friedrich method and compactness arguments. Furthermore, we also obtain the optimal decay rate for the Korteweg system in $L^2$ type Besov spaces.

Very weak solutions of wave equation for Landau Hamiltonian with irregular electromagnetic field

In this paper we study the Cauchy problem for the Landau Hamiltonian wave equation, with time dependent irregular (distributional) electromagnetic field and similarly irregular velocity. For such equations, we describe the notion of a `very weak solution' adapted to the type of solutions that exist for regular coefficients. The construction is based on considering Friedrichs-type mollifier of the coefficients and corresponding classical solutions, and their quantitative behaviour in the regularising parameter. We show that even for distributional coefficients, the Cauchy problem does have a ve…

基于交替正交投影的椭圆柯西问题迭代格式的新收敛性证明

A new convergence proof for an iterative scheme for elliptic Cauchy problems based on alternating orthogonal projections

针对边界部分给定条件的不适定椭圆柯西问题，研究基于交替正交投影的迭代方法，给出其收敛性证明及收敛速率，同时指出该速率估计会退化。

渐近 Anti-de Sitter 空间的几何反射边界条件

Geometric reflective boundary conditions for asymptotically Anti-de Sitter spaces

本文提出一族涉及时形共形边界共形类与应力-能量张量的几何反射边界条件，证明四维负宇宙学常数真空 Einstein 方程初边值问题的局部适定性。

On the Scattering of Waves inside Charged Spherically Symmetric Black Holes

In this paper we show that there is a breakdown of scattering between the event horizon (or the Cauchy horizon) and an intermediate Cauchy hypersurface in the dynamic interior of a Reissner-Nordström-like black hole. More precisely, we show that the trace operators and their analytic counterparts, the inverse wave operators, do not have bounded inverses, even though these operators themselves are bounded. This result holds for the natural energy given by the energy-momentum tensor of the wave equation using the timelike vector field of the Regge-Wheeler variable, which asymptotically becomes n…

使用能量差代价泛函重新探讨 Bernoulli 自由边界问题最优解的存在性证明

Revisited for existence proof of optimal solution in Bernoulli free boundary problem using an energy-gap cost functional

本文纠正了文献中关于 Bernoulli 自由边界问题最优解存在性证明的勘误，指出解序列有界性不能直接用 Cauchy-Schwarz 不等式估计，并改用 Poincaré-Friedrichs 不等式完成了证明。

广义相对论中适定的几何边界数据，III：共形平均曲率边界数据

Well-posed geometric boundary data in General Relativity, III: Conformal-mean curvature boundary data

研究真空爱因斯坦方程初边值问题在共形平均曲率边界条件下的局部适定性，通过线性化分析和Holmgren型唯一性定理，证明解空间在光滑函数中稠密。